The above definition specifically treats the real case, but it can be similarly defined for the complex case in the obvious way.
For the complex case, it is known that \(2M\)  measurements are necessary for almost injectivity [1]}, and that \(2M\)  generic measurements suffice [2]}; this is the factor-of-2 savings mentioned above.
For the real case, it is also known how many measurements are necessary and generically sufficient for almost injectivity: \(M+1\)   [2]}.
Like the complex case, this is also a factor-of-2 savings from the injectivity requirement: \(2M-1\) .
This requirement for injectivity in the real case follows from the following result from  [2]}, which we prove here because the proof is short and inspires the remainder of this section:
